Results 1  10
of
17
The quantitative structure of exponential time
 Complexity Theory Retrospective II
, 1997
"... ..."
(Show Context)
Equivalence of Measures of Complexity Classes
"... The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases ..."
Abstract

Cited by 73 (22 self)
 Add to MetaCart
The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomialtime, truthtable reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the cointoss probability measure given by the sequence ~ fi. (2) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...
The Complexity and Distribution of Hard Problems
 SIAM JOURNAL ON COMPUTING
, 1993
"... Measuretheoretic aspects of the P m reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in ..."
Abstract

Cited by 49 (18 self)
 Add to MetaCart
(Show Context)
Measuretheoretic aspects of the P m reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are P m  hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bound says that the P m hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E. It follows that the P m complete languages for E form a measure 0 subset of E (and similarly in E 2 ). This latter fact is seen to be a special case of a more general theorem, namely, that every P m degree (e.g., the degree of all P m complete languages for NP) has measure 0 in E and in E 2 .
ResourceBounded Measure and Randomness
"... We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than ..."
Abstract

Cited by 42 (6 self)
 Add to MetaCart
We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than timebounded randomness but which suffice for many of the applications in complexity theory.
Resource Bounded Randomness and Weakly Complete Problems
 Theoretical Computer Science
, 1994
"... We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([7, 8]). We concentrate on n c  randomness (c 2) which corresponds to the polynomial time bounded (p) measure of Lutz, and which is adequate for studying the internal and quantitative s ..."
Abstract

Cited by 36 (7 self)
 Add to MetaCart
(Show Context)
We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([7, 8]). We concentrate on n c  randomness (c 2) which corresponds to the polynomial time bounded (p) measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTIME(2 lin ). However we will also comment on E2 = DTIME(2 pol ) and its corresponding (p2 ) measure. First we show that the class of n c random sets has pmeasure 1. This provides a new, simplified approach to pmeasure 1results. Next we compare randomness with genericity (in the sense of [2, 3]) and we show that n c+1  random sets are n c generic, whereas the converse fails. From the former we conclude that n c random sets are not pbttcomplete for E. Our technical main results describe the distribution of the n c random sets under pmreducibility. We show that every n c random set in E has n k random predecessors in E for any k 1, whereas the amou...
ResourceBounded Baire Category: A Stronger Approach
 Proceedings of the Tenth Annual IEEE Conference on Structure in Complexity Theory
, 1996
"... This paper introduces a new definition of resourcebounded Baire category in the style of Lutz. This definition gives an almostall/almostnone theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new definition than in previous definitions. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
This paper introduces a new definition of resourcebounded Baire category in the style of Lutz. This definition gives an almostall/almostnone theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new definition than in previous definitions. For example, almost no sets in EXP are EXPcomplete, and NP is PFmeager unless NP = EXP. It is also seen under the new definition that no recrandom set can be (recursively) ttreducible to any PFgeneric set. We weaken our definition by putting arbitrary bounds on the length of extension strategies, obtaining a spectrum of different theories of Baire Category that includes Lutz's original definition. 1
Weakly Useful Sequences
, 2004
"... An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a nonmeasure 0 subset of the set of dec ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a nonmeasure 0 subset of the set of decidable sequences are Turing reducible to x. Juedes,
Almost Complete Sets
, 2000
"... . We show that there is a set which is almost complete but not complete under polynomialtime manyone (pm) reductions for the class E of sets computable in deterministic time 2 lin . Here a set A in a complexity class C is almost complete for C under some reducibility r if the class of the p ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
. We show that there is a set which is almost complete but not complete under polynomialtime manyone (pm) reductions for the class E of sets computable in deterministic time 2 lin . Here a set A in a complexity class C is almost complete for C under some reducibility r if the class of the problems in C which do not rreduce to A has measure 0 in C in the sense of Lutz's resourcebounded measure theory. We also show that the almost complete sets for E under polynomialtime bounded oneone lengthincreasing reductions and truthtable reductions of norm 1 coincide with the almost pmcomplete sets for E. Moreover, we obtain similar results for the class EXP of sets computable in deterministic time 2 poly . 1 Introduction Lutz [15] introduced measure concepts for the standard deterministic time and space complexity classes which contain the class E of sets computable in deterministic time 2 lin . These measure concepts have been used for investigating quantitative aspe...